Noise Texture Stretching on Side of Cube
You need to connect your UV map.

Image textures use generated coordinates by default - this makes your texture stretched.
Noise texture works fine with generated coordinates, so it looks okay in your case.

Solution:
1) Add a Texture coordinate node and a Mapping node.
2) Connect your UV map output.
3) Done  



Comparison between different variants
Left - properly connected UV map
Center - texture without UV map connected (clearly distorted)
Right - Procedural textures work just fine